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Classification systems are often deployed in resource-constrained settings where labels must be assigned to inputs on a budget of time, memory, etc. Budgeted, sequential classifiers (BSCs) address these scenarios by processing inputs through a sequence of partial feature acquisition and evaluation steps with early-exit options. This allows for an efficient evaluation of inputs that prevents unneeded feature acquisition. To approximate an intractable combinatorial problem, current approaches to budgeted classification rely on well-behaved loss functions that account for two primary objectives (processing cost and error). These approaches offer improved efficiency over traditional classifiers but are limited by analytic constraints in formulation and do not manage additional performance objectives. Notably, such methods do not explicitly account for an important aspect of real-time detection systems -- the fraction of "accepted" predictions satisfying a confidence criterion imposed by a risk-averse monitor. We propose a problem-specific genetic algorithm to build budgeted, sequential classifiers with confidence-based reject options. Three objectives -- accuracy, processing time/cost, and coverage -- are considered. The algorithm emphasizes Pareto efficiency while accounting for a notion of aggregate performance via a unique scalarization. Experiments show our method can quickly find globally Pareto optimal solutions in very large search spaces and is competitive with existing approaches while offering advantages for selective, budgeted deployment scenarios.